a scalability analysis over datasets with millions of events showed that other CTES models took up to 24 hours training times, whereas the times for IMTPP were under 5 hours.
Ongoing Work and Challenges.
As an extension to IMTPP, we addressed the problem of imputing missing events in a sequence. Specifically, we sampled missing events using the posterior MTPP and evaluated over synthetically deleted events in a sequence. The results, including qualitative analysis, showed that the new approach outperforms other baselines and opens up new applications of neural models and CTES. This extension is under review in a reputed journal. A major challenge with imputations is to estimate the total number of missing events between each observed event.
Limited Training Data
The problem of limited training data is ubiquitous in all real-world applications ranging from recommender systems [1]}, vision [2]}, spatial models [3]} etc. In contrast to incomplete data, defined in Section REF , we regard limited data as the problem of data-scarcity i.e. the available data is assumed to be complete but the volume is insufficient to effectively train a deep neural network.
Proposed Solution.
Transfer learning (TL) has long been proposed as a feasible solution to overcome limited data problems [4]}. Accordingly, in [5]}, we present a transfer approach for training neural MTPP on a data-rich dataset and fine-tune the model parameters on a data-scarce dataset. In detail, we consider the problem of mobility prediction, wherein we have spatial trajectories of users across different regions and regions with data-rich mobility sequence as a source region and the target region with scarce mobility data.
The proposed model REFORMD(Reusable Flows for Mobility Data), learns the spatial and temporal distribution of a user trajectory using normalizing flows(NFs) [6]}. We make the trained NFs invariant to a region, by restricting our model to learn the distribution of inter-event time intervals, \(\Delta _{t,\bullet }\) , and spatial distances, \(\Delta _{d,\bullet }\) . Since these features are independent of the underlying region, the trained NFs can be easily fine-tuned for any mobility sequence. Moreover, we use a log-normal flow to model both distributions e.g. for time, the generative distribution for future events is learned as:
\(p_t(\Delta _{t, k+1} | s_k) = \texttt {LogNormal} \big (\mu _t(s_k), \sigma ^2_t(s_k)\big ),\) 
with \([\mu _t(s_k), \sigma ^2_t(s_k)] = [W_{1} s_k + b_1, W_{2} s_k + b_2]\)  denote the mean and variance of the time distribution. \(s_k\)  is the output of neural MTPP, \(W_{\bullet }\)  and \(b_{\bullet }\)  are trainable parameters. We sample the time of future events as \(\Delta _{t, k+1} \sim p_t\)  and a similar procedure is followed for spatial flows. All model parameters are trained by maximizing the location recommendation accuracy and the likelihood of time and distance of events, Moreover, we use a standard procedure to fine-tune the model parameters on the sequences from the target region.
Results.
We evaluated our model across eight mobility datasets from the US and Japan and our observations were:
for location recommendation and event-time prediction, REFORMD outperformed other MTPP models by up to 20% and 23% respectively

It also demonstrated better and faster convergence on target dataset than other approaches, and

even in the absence of spatial flows, we outperform other methods by 3% for item recommendation and 14% for time prediction across four datasets from Amazon.
Ongoing Work and Challenges.
A crucial drawback of our proposed model is its standard transfer learning procedure, whereas, modern neural models deploy a meta-learning [7]} procedure to transfer model parameters. Therefore as future work, we plan to combine meta-learning with NF to design robust MTPP models.

Latent Features
The latent features that play a crucial role in CTES are inter-event and inter-sequence relationships. [8]} show that these relationships imitate an information diffusion process e.g. social preferences, community formations etc. and can be captured using MTPP models. Accordingly, we address the problem of community detection in the absence of the social network of a user. Specifically, we assign communities to users in a network based on their mobility preferences.
Proposed Solution.
In our work [9]} with Ankita, a former Ph.D. student, we devise a spatial-temporal point process method to learn the diffusion process to assign communities to users in the network. In detail, we train a self-exciting MTPP for each user with a common excitation matrix. Later, we assign a community to a user based on their personal MTPP and the shared matrix. We learn the latent community of users and our model parameters using stochastic variational inference.
Results and Challenges.
The results across two spatial mobility datasets show that our model achieves improvements of up to 27% in location prediction and 8% for community prediction in comparison to other neural models. Though we use certain heuristics to estimate the user community, evaluating it in the absence of true labels is still an open problem.

Ongoing Work
Here we highlight the ongoing works that are under review.
Time Series as Graphs. In this work, we represent a time series as a graph with weighted temporal edges. Later, we learn the dynamics of sequences using graph neural networks. Moreover, we overcome data scarcity using a meta-learning algorithm.
Retrieving CTES. Due to the disparate nature of CTES with events containing marks and if applicable, spatial features, the problem of sequences retrieval has been left undressed by the past literature. In this work, we propose a self-attention MTPP model for retrieving a similar sequence from a corpus, given a query sequence.
Time Series and Databases. In [10]} with Garima, a former Ph.D. student, we learn the embedding of entities in temporally evolving relational databases.


Conclusion
In this paper, we present the research directions, possible data-related problems, and solutions for learning neural models on continuous time sequences. In addition, we highlighted the ongoing works that are under review. As possible progress till October, we plan to get better results for meta-learning-based training of MTPP and notifications for the papers under review.
